Single-quanta interferometry: which-way versus which-phase information stored in an ancillary quantum system

TL;DR

This work analyzes how information about a single quantum in a two-path interferometer is distributed between which-path information stored in an ancillary quantum system (AQS) and which-phase information obtainable via that same AQS. The authors adopt Shannon entropy and define two guessing games (WWGG and WPGG), deriving that the entropic which-phase information $H(\Phi|\mathcal D)-H(\Phi|\mathcal D,\mathcal E)$ is bounded above by the which-way accessible information $I({\cal W}:{\cal M})$, with equality in the symmetric, pure-state case where $p=0.5$; in general $H(\Phi|\mathcal D)-H(\Phi|\mathcal D,\mathcal E) \le I({\cal W}:{\cal M})$. They provide explicit formulas: for the post-interaction overlaps $r_w=\langle w_0|w_1\rangle$ and $r_d=|\langle d_0^{\phi_0}|d_1^{\phi_0}\rangle| = \sqrt{(2p-1)^2+4 r_w^2 p(1-p)\sin^2(\gamma+\phi_0)}$, the two quantities relate as $H(\Phi|\mathcal D)-H(\Phi|\mathcal D,\mathcal E)=I_A(r_d,0.5)$; equality occurs for $p=0,0.5,1$ or $r_w=1$ and the symmetric case yields equality. An experimental proposal using a modified Mach-Zehnder interferometer with a cavity QED ancilla demonstrates feasibility. The results illuminate a sharp entropic trade-off between path information and phase information in binary interferometers and propose a practical route to verify it.

Abstract

In interferometers, the more information about the quanta's path available in an ancillary quantum system (AQS), the less visibility the interference has. By use of Shannon entropy, we try to compare the amount of which-phase information with the amount of which-way information stored in the AQS of two-path interferometers with symmetric beam merging. We show that the former is lower than or equal the latter if the bipartite system of the single-quanta and the AQS is initially prepared in a pure state and the interaction between the two parts is unitary. Especially when there exists symmetry, the equality holds. No which-way information is obtained by the measurement that we use for extracting the which-phase information and vice versa. In order to verify the results experimentally, we propose assembling a new single-photon interferometer.

Figures (2)

• Figure 1: (a) Scheme of an ordinary two-path interferometer. A beam splitter (BS) makes a linear combination of the two paths. Then a phase difference between the arms is made by a phase shifter (PS) and a beam merger (BM) combines the paths. Finally, one of the two detectors $D_0$ or $D_1$ detects the single quanta. (b) An ancillary quantum system (AQS) interacts with the single quanta within the interferometer. If the system is correlated with the quanta's spatial degree of freedom after this interaction, it stores some amount of information about the path of the single quanta.
• Figure 2: A single photon is sent toward a beam splitter (BS). In the lower arm, it interacts with a three-level atom trapped in a high-finesse cavity -- the resonance frequency of the cavity equals the transition frequency between $|1\rangle$ and $|e\rangle$ and is around the frequency of the single photon. The polarization beam splitter (PBS) and the wave plates are used for directing the photon in the path. Then the photon passes through a phase shifter (PS) and experiences a $\phi$ phase shift. A 50:50 BS merges the two arms. Finally, $D_0$ or $D_1$ detects the single photon.